Question

(1 point) A normal distribution with mean and variance o is independently sampled three times, yielding values x1, x2, and X3. Consider the three estimators û = X1 + 5x2 A2 = x - x2 + x3, and Find the expected value of each estimator (type mu for and sigma foro): E) EG) E) = Which estimator(s) are biased and which are unbiased? Estimator : ? Estimator 2: ? Estimators: ? Find the variance of each estimator (type mu for u and sigma for o): var(A) = var(a) = var(A) = Which estimator is the most efficient? ?

Answer 1

23 ů = x - H2+2 3 and bolun 3.Xand als are independently sampled from N(462) ů, = 2, +522 a = 12 + 1 2 + 3 a :. Expented value for each estimator Ell,)E (29,7542) = E(21] +5 EC42) = 4+54 = 64 "E(â) = 64 Elân) - E (3,-22+43) = E(2) -E(42)+E (713) = 4-1+4 = 4 Elîz) = 4 3 Elîs) = E(52,75 223) = 5 E (40) + 5 E (2x) + - (213) u+tu+ 3 4 & M = 4 ..Elâz) = 4 Rt 5 I S .. · Which estimators are bised and which are unbiased ..Elê ) = 64 which is bised for us Elû, ) - bised for u. :E(ů) = 4 which is unbised for us : Eld) = unbised Zov 4. Elứa) = 4 + which is unbised 30 4: Elûg) - unbised Bor 4, variance of each estimator, Here H, ,312 and He crre independent so all covawance term are o ... Varlû, ) = Var (2, +5+42) = Var (20,2 +25. Var (242) = 0 °+250?=2607 . Var ( , ) = 2602 var(ú) = VOY (20,- Hl2 + H2) = Vay (240) + var(246) + Var (23) -0%+04+02-302 var (et) = 302

var (g) = Var (fa, + b2 +3 13 ) Ivar (21) + L var (22) 9 کی + Vay (1) 25 25 25 2. ㅗ 25 + Lo2 Le + 62 25 25 U 02 25 . Vo Var (3) = 1 2 . 25